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There is growing interest in the design of pension annuities that insureagainst idiosyncratic longevity risk while pooling and sharing systematic risk.This is partially motivated by the desire to reduce capital and reserverequirements while retaining the value of mortality credits; see for examplePiggott, Valdez and Detzel (2005) or Donnelly, Guillen and Nielsen (2014). Inthis paper we generalize the natural retirement income tontine introduced byMilevsky and Salisbury (2015) by combining heterogeneous cohorts into one pool.We engineer this scheme by allocating tontine shares at either a premium or adiscount to par based on both the age of the investor and the amount theyinvest. For example, a 55 year-old allocating $\$10,000$ to the tontine mightbe told to pay $\$$200 per share and receive 50 shares, while a 75 year-oldallocating $\$8,000$ might pay $\$$40 per share and receive 200 shares. Theywould all be mixed together into the same tontine pool and each tontine sharewould have equal income rights. The current paper addresses existence anduniqueness issues and discusses the conditions under which this scheme can beconstructed equitably -- which is distinct from fairly -- even though it isn'toptimal for any cohort. As such, this also gives us the opportunity to compareand contrast various pooling schemes that have been proposed in the literatureand to differentiate between arrangements that are socially equitable, vs.actuarially fair vs. economically optimal.

Equitable retirement income tontines: Mixing cohorts without discriminating